Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-7890976x-8534500108\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-7890976xz^2-8534500108z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-639169083x-6219733071510\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1622, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1622, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 25872 \) | = | $2^{4} \cdot 3 \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $931306984194048$ | = | $2^{14} \cdot 3 \cdot 7^{6} \cdot 11^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{112763292123580561}{1932612} \) | = | $2^{-2} \cdot 3^{-1} \cdot 11^{-5} \cdot 179^{3} \cdot 2699^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4139555214841496200851841473$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.74785326639654765811527565412$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0637869122465706$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.831884736035459$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.090077613162592089293330080395$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.5038806581296044646665040198 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.503880658 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.090078 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 4.503880658\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 864000 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.6.0.4 |
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 6929 & 6580 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 4621 & 2646 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9221 & 20 \\ 9220 & 21 \end{array}\right),\left(\begin{array}{rr} 5286 & 2653 \\ 5635 & 7736 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 9000 & 8891 \end{array}\right),\left(\begin{array}{rr} 3959 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1926 & 2653 \\ 875 & 7736 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3697 & 2660 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$3269984256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \) |
| $5$ | good | $2$ | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 25872cs
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66c3, its twist by $28$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.413952.2 | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.98000.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.0.186606965293056.58 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.793808535953664.29 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.11389585284000000.207 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.2.17643316320000000000.2 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.4.72665756964764081922245052441600000000000000000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 |
|---|---|---|---|---|---|
| Reduction type | add | split | ord | add | nonsplit |
| $\lambda$-invariant(s) | - | 1 | 2 | - | 0 |
| $\mu$-invariant(s) | - | 0 | 1 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.